LOOSE HAMILTONIAN CYCLES FORCED BY LARGE pk ´ 2q-DEGREE
– SHARP VERSION –
arXiv:1705.03707v1 [math.CO] 10 May 2017
JOSEFRAN DE OLIVEIRA BASTOS, GUILHERME OLIVEIRA MOTA, MATHIAS SCHACHT,
JAKOB SCHNITZER, AND FABIAN SCHULENBURG
Abstract. We prove for all k ě 4 and 1 ď ℓ ă k{2 the sharp minimum pk ´ 2q-degree
bound for a k-uniform hypergraph H on n vertices to contain a Hamiltonian ℓ-cycle if
k ´ ℓ divides n and n is sufficiently large. This extends a result of Han and Zhao for
3-uniform hypegraphs.
§1. Introduction
Given k ě 2, a k-uniform hypergraph H is a pair pV, Eq with vertex set V and edge
set E Ď V pkq being a subset of all k-element subsets of V . Given a k-uniform hypergraph
H “ pV, Eq and a subset S P V psq , we denote by dpSq the number of edges in E containing S
and we denote by NpSq the pk ´ sq-element sets T P V pk´sq such that T Ÿ S P E, so
dpSq “ |NpSq|. The minimum s-degree of H is denoted by δs pHq and it is defined as the
minimum of dpSq over all sets S P V psq .
We say that a k-uniform hypergraph C is an ℓ-cycle if there exists a cyclic ordering of
its vertices such that every edge of C is composed of k consecutive vertices, two (vertexwise) consecutive edges share exactly ℓ vertices, and every vertex is contained in an edge.
Moreover, if the ordering is not cyclic, then C is an ℓ-path and we say that the first
and last ℓ vertices are the ends of the path. The problem of finding minimum degree
conditions that ensure the existence of Hamiltonian cycles, i.e. cycles that contain all
vertices of a given hypergraph, has been extensively studied over the last years (see, e.g.,
the surveys [11, 14]). Katona and Kierstead [7] started the study of this problem, posing
a conjecture that was confirmed by Rödl, Ruciński, and Szemerédi [12, 13], who proved
the following result: For every k ě 3, if H is a k-uniform n-vertex hypergraph with
δk´1 pHq ě p1{2 ` op1qqn, then H contains a Hamiltonian pk ´ 1q-cycle. Kühn and Osthus
proved that 3-uniform hypergraphs H with δ2 pHq ě p1{4 ` op1qqn contain a Hamiltonian
2010 Mathematics Subject Classification. 05C65 (primary), 05C45 (secondary).
Key words and phrases. hypergraphs, Hamiltonian cycles, degree conditions.
The first author was supported by CAPES. The second author was supported by FAPESP (Proc.
2013/11431-2 and 2013/20733-2) and CNPq (Proc. 477203/2012-4 and 456792/2014-7). The cooperation
was supported by a joint CAPES/DAAD PROBRAL (Proc. 430/15).
1
2
J. DE O. BASTOS, G. O. MOTA, M. SCHACHT, J. SCHNITZER, AND F. SCHULENBURG
1-cycle [10], and Hàn and Schacht [4] (see also [8]) generalized this result to arbitrary k
and ℓ-cycles with 1 ď ℓ ă k{2. In [9], Kühn, Mycroft, and Osthus generalized this result
to 1 ď ℓ ă k, settling the problem of the existence of Hamiltonian ℓ-cycles in k-uniform
hypergraphs with large minimum pk ´ 1q-degree. In Theorem 1 below (see [1, 2]) we have
minimum pk ´ 2q-degree conditions that ensure the existence of Hamiltonian ℓ-cycles for
1 ď ℓ ă k{2.
Theorem 1. For all integers k ě 3 and 1 ď ℓ ă k{2 and every γ ą 0 there exists
an n0 such that every k-uniform hypergraph H “ pV, Eq on |V | “ n ě n0 vertices with
n P pk ´ ℓqN and
δk´2 pHq ě
contains a Hamiltonian ℓ-cycle.
ˆ
4pk ´ ℓq ´ 1
`γ
4pk ´ ℓq2
˙ˆ ˙
n
2
The minimum degree condition in Theorem 1 is asymptotically optimal as the following
well-known example confirms. The construction of the example varies slightly depending
on whether n is an odd or an even multiple of k ´ ℓ. We first consider the case that
n “ p2m ` 1qpk ´ ℓq for some integer m. Let Xk,ℓ pnq “ pV, Eq be a k-uniform hypergraph
on n vertices such that anYedge belongs
to E if and only if it contains at least one vertex
]
n
from A Ă V , where |A| “ 2pk´ℓq . It is easy to see that Xk,ℓ pnq contains no Hamiltonian
ℓ-cycle, as it would have to contain
n
k´ℓ
edges and each vertex in A is contained in at
most two of them. Indeed any maximal ℓ-cycle includes all but k ´ ℓ vertices and adding
any additional edge to the hypergraph would imply a Hamiltonian ℓ-cycle. Let us now
consider the case that n “ 2mpk ´ ℓq for some integer m. Similarly, let Xk,ℓ pnq “ pV, Eq
be a k-uniform hypergraph on n vertices that contains all edges incident to A Ă V , where
|A| “
n
2pk´ℓq
´ 1. Additionally, fix some ℓ ` 1 vertices of B “ V r A and let Xk,ℓ pnq
contain all edges on B that contain all of these vertices, i.e., an pℓ ` 1q-star. Again, of
the
n
k´ℓ
edges that a Hamiltonian ℓ-cycle would have to contain, at most
n
k´ℓ
´ 2 can be
incident to A. So two edges would have to be completely contained in B and be disjoint
or intersect in exactly ℓ vertices, which is impossible since the induced subhypergraph on
B only contains an pℓ ` 1q-star. Note that for the minimum pk ´ 2q-degree the pℓ ` 1q-star
on B is only relevant if ℓ “ 1, in which case this star increases the minimum pk ´ 2q-degree
by one.
In [6], Han and Zhao proved the exact version of Theorem 1 when k “ 3, i.e., they
obtained a sharp bound for δ1 pHq. We extend this result to k-uniform hypergraphs.
Theorem 2 (Main Result). For all integers k ě 4 and 1 ď ℓ ă k{2 there exists n0 such
that every k-uniform hypergraph H “ pV, Eq on |V | “ n ě n0 vertices with n P pk ´ ℓqN
LOOSE HAMILTONIAN CYCLES FORCED BY pk ´ 2q-DEGREE
3
and
δk´2 pHq ą δk´2 pXk,ℓ pnqq
(1)
contains a Hamiltonian ℓ-cycle. In particular, if
ˆ ˙
4pk ´ ℓq ´ 1 n
δk´2 pHq ě
,
2
4pk ´ ℓq2
then H contains a Hamiltonian ℓ-cycle.
The following notion of extremality is motivated by the hypergraph Xk,ℓ pnq. A k-uniform
hypergraphQH “ pV, Eq
U is called
Y pℓ, ξq-extremal
] if there exists a partition `V ˘“ A Ÿ B such
2pk´ℓq´1
n
that |A| “ 2pk´ℓq
´ 1 , |B| “ 2pk´ℓq n ` 1 and epBq “ |E X B pkq | ď ξ nk . We say that
A Ÿ B is an pℓ, ξq-extremal partition of V . Theorem 2 follows easily from the next two
results, the so-called extremal case (see Theorem 4 below) and the non-extremal case (see
Theorem 3).
Theorem 3 (Non-extremal Case). For any 0 ă ξ ă 1 and all integers k ě 4 and 1 ď ℓ ă
k{2, there exists γ ą 0 such that the following holds for sufficiently large n. Suppose H is
a k-uniform hypergraph on n vertices with n P pk ´ ℓqN such that H is not pℓ, ξq-extremal
and
ˆ
4pk ´ ℓq ´ 1
δk´2 pHq ě
´γ
4pk ´ ℓq2
Then H contains a Hamiltonian ℓ-cycle.
˙ˆ ˙
n
.
2
The non-extremal case was the main result of [1].
Theorem 4 (Extremal Case). For any integers k ě 3 and 1 ď ℓ ă k{2, there exists ξ ą 0
such that the following holds for sufficiently large n. Suppose H is a k-uniform hypergraph
on n vertices with n P pk ´ ℓqN such that H is pℓ, ξq-extremal and
δk´2 pHq ą δk´2 pXk,ℓ q.
Then H contains a Hamiltonian ℓ-cycle.
In Section 2 we give an overview of the proof of Theorem 4 and state Lemma 5, the
main result required for the proof. In Section 3 we first prove some auxiliary lemmas and
then we prove Lemma 5.
§2. Overview
Let H “ pV, Eq be a k-uniform hypergraph and let X, Y Ă V be disjoint subsets. Given
a vertex set L Ă V we denote by dpL, X piq Y pjq q the number of edges of the form L Y I Y J,
4
J. DE O. BASTOS, G. O. MOTA, M. SCHACHT, J. SCHNITZER, AND F. SCHULENBURG
where I P X piq , J P Y pjq , and |L| ` i ` j “ k. We allow for Y pjq to be omitted when j is
zero and write dpv, , X piqY pjq q for dptvu, , X piq Y pjq q.
The proof of Theorem 4 follows ideas from [5], where a corresponding result with a pk´1q-
degree condition is proved. Let H “ pV, Eq be an extremal hypergraph satisfying (1). We
first construct an ℓ-path Q in H (see Lemma 5 below) with ends L0 and L1 such that there
is a partition A˚ Ÿ B˚ of pV r Qq Y L0 Y L1 composed only of “typical” vertices (see (ii )
and (iii ) below). The set A˚ Y B˚ is suitable for an application of Lemma 6 below, which
ensures the existence of an ℓ-path Q1 on A˚ Y B˚ with L0 and L1 as ends. Note that the
existence of a Hamiltonian ℓ-cycle in H is guaranteed by Q and Q1 . So, in order to prove
Theorem 4, we only need to prove the following lemma.
Lemma 5 (Main Lemma). For any ̺ ą 0 and all integers k ě 3 and 1 ď ℓ ă k{2, there
exists a positive ξ such that the following holds for sufficiently large n P pk ´ ℓqN. Suppose
that H “ pV, Eq is an pℓ, ξq-extremal k-uniform hypergraph on n vertices and
δk´2 pHq ą δk´2 pXk,ℓ pnqq.
Then there exists a non-empty ℓ-path Q in H with ends L0 and L1 and a partition A˚ ŸB˚ “
pV r Qq Y L0 Y L1 where L0 , L1 Ă B˚ such that the following hold:
(i ) |B˚ | “ p2k ´ 2ℓ ´ 1q|A˚ | ` ℓ,
` ˚ |˘
pk´1q
(ii ) dpv, B˚
q ě p1 ´ ̺q |B
for any vertex v P A˚ ,
k´1
` ˚ |˘
p1q pk´2q
(iii ) dpv, A˚ B˚
q ě p1 ´ ̺q|A˚ | |B
for any vertex v P B˚ ,
k´2
` |B˚ | ˘
p1q pk´ℓ´1q
p1q pk´ℓ´1q
(iv ) dpL0 , A˚ B˚
q, dpL1 , A˚ B˚
q ě p1 ´ ̺q|A˚ | k´ℓ´1
.
The next result, which we will use to conclude the proof of Theorem 4, was obtained by
Han and Zhao (see [5, Lemma 3.10]).
Lemma 6. For any integers k ě 3 and 1 ď ℓ ă k{2 there exists ̺ ą 0 such that
the following holds. If H is a sufficiently large k-uniform hypergraph with a partition
V pHq “ A˚ Ÿ B˚ and there exist two disjoint ℓ-sets L0 , L1 Ă B˚ such that (i )–(iv ) hold,
then H contains a Hamiltonian ℓ-path Q1 with L0 and L1 as ends.
§3. Proof of the Main Lemma
We will start this section by describing the setup for the proof, which will be fixed
for the rest of the paper. Then we will prove some auxiliary lemmas and finally prove
Lemma 5. Let ̺ ą 0 and integers k ě 3 and 1 ď ℓ ă k{2 be given. Fix constants
1 1
, , ̺ " δ " ε " ε1 " ϑ " ξ.
k ℓ
LOOSE HAMILTONIAN CYCLES FORCED BY pk ´ 2q-DEGREE
5
Let n P pk ´ℓqN be sufficiently large and let H be an pℓ, ξq-extremal k-uniform hypergraph
on n vertices that satisfies the pk ´ 2q-degree condition
δk´2 pHq ą δk´2 pXk,ℓ pnqq.
Let A Ÿ B “ V pHq be a minimal extremal partition of V pHq, i.e. a partition satisfying
R
V
ˆ ˙
n
n
a “ |A| “
,
(2)
´ 1, b “ |B| “ n ´ a, and epBq ď ξ
k
2pk ´ ℓq
which minimises epBq. Recall that the extremal example Xk,ℓ pnq implies
ˆ ˙
a
` apb ´ k ` 2q.
(3)
δk´2 pHq ą
2
` ˘
Since epBq ď ξ nk , we expect most vertices v P B to have low degree dpv, B pk´1q q into B.
Also, most v P A must have high degree dpv, B pk´1qq into B such that the degree condition
for pk ´ 2q-sets in B can be satisfied. Thus, we define the sets Aε and Bε to consist of
vertices of high respectively low degree into B by
˙*
"
ˆ
|B|
pk´1q
,
Aε “ v P V : dpv, B
q ě p1 ´ εq
k´1
˙*
"
ˆ
|B|
pk´1q
,
Bε “ v P V : dpv, B
qďε
k´1
and set Vε “ V r pAε Y Bε q. We will write aε “ |Aε |, bε “ |Bε |, and vε “ |Vε |. It follows
from these definitions that
if A X Bε ‰ ∅,
then B Ă Bε ,
while otherwise A Ă Aε .
(4)
For the first inclusion, consider a vertex v P AXBε and a vertex w P B rBε . Exchanging v
and w would create a minimal partition with fewer edges in epBq, a contradiction to the
minimality of the extremal partition. The other inclusion is similarly implied by the
minimality.
Actually, as we shall show below, the sets Aε and Bε are not too different from A and
B respectively:
|A r Aε |, |B r Bε |, |Aε r A|, |Bε r B| ď ϑb and |Vε | ď 2ϑb.
(5)
Note that by the minimum pk ´ 2q-degree
˙
˙
ˆ
˙
ˆ
ˆ ˙ˆ
ÿ
b
b
b
a
δk´2 pHq ď
pk ´ 1q ă
dpSq.
`a
k´2
k´1
2 k´2
SPB pk´2q
` b ˘
Every vertex v P |A r Aε | satisfies dpv, B pk´1q q ă p1 ´ εq k´1
, so we have
˙
ˆ
ˆ ˙
˙
˙
ˆ
ˆ ˙ˆ
ÿ
b
k
b
b
a
pk ´ 1q
´ |A r Aε |εb
pk ´ 1q ` epBq
`a
dpSq ď
k´1
2
k´1
2 k´2
pk´2q
SPB
6
J. DE O. BASTOS, G. O. MOTA, M. SCHACHT, J. SCHNITZER, AND F. SCHULENBURG
Consequently |A r Aε | ď ϑb, as epBq ă ξ
`n˘
k
and ξ ! ϑ, ε.
Moreover, |B r Bε | ď ϑb holds as a high number of vertices in B r Bε would contradict
`˘
epBq ă ξ kb . The other three inequalities (5) follow from the already shown ones, for
example for |Aε r A| ă ϑb observe that
Aε r A “ Aε X B Ă B r Bε .
Although the vertices in Bε were defined by their low degree into B, they also have low
degree into the set Bε itself; for any v P Bε we get
˙
˙
ˆ
˙
ˆ
ˆ
|Bε |
b
|Bε | ´ 1
k´1
pk´1q
pk´1q
.
` ϑb|Bε |
ă 2ε
ďε
dpv, Bε
q ď dpv, B
q ` |Bε r B|
k´1
k´1
k´2
Since we are interested in ℓ-paths, the degree of ℓ-tuples in Bε will be of interest, which
motivates the following definition. An ℓ-set L Ă Bε is called ε-typical if
˙
ˆ
|B|
pk´ℓq
.
dpL, B
qďε
k´ℓ
If L is not ε-typical, then it is called ε-atypical. Indeed, most ℓ-sets in Bε are ε-typical;
denote by x the number of ε-atypical sets in Bε . We have
` |B| ˘
˙
ˆ
ˆ ˙
x ¨ ε k´ℓ
n
k
1 |Bε |
`k ˘
.
` ϑ|B| , implying x ď ε
ď epB Y Bε q ď ξ
ℓ
k
ℓ
(6)
Lemma 7. The following holds for any Bεpmq -set M if m ď k ´ 2.
˙
ˆ
k´m
|Bε | ´ m
pk´mq
p1q pk´m´1q
.
dpM, Bε
q ě p1 ´ δq |Aε |
dpM, Aε Bε
q`
2
k´m´1
In particular, the following holds for any ε-typical B pℓq -set L.
˙
ˆ
|Bε | ´ ℓ
p1q pk´ℓ´1q
.
dpL, Aε Bε
q ě p1 ´ 2δq|Aε |
k´ℓ´1
In the proof of the main lemma we will connect two ε-typical sets only using vertices
that are unused so far. Even more, we want to connect two ε-typical sets using exactly
one vertex from A. The following corollary of Lemma 7 allows us to do this.
Corollary 8. Let L and L1 be two disjoint ε-typical sets in Bε and U Ă V with |U| ď εn.
Then the following holds.
(a ) There exists an ℓ-path disjoint from U of size two with ends L and L1 that contains
exactly one vertex from Aε .
(b ) There exist a P Aε r U and a set pk ´ ℓ ´ 1q-set C Ă Bε r U such that L Y a Y C
is an edge in H and every ℓ-subset of C is ε-typical.
LOOSE HAMILTONIAN CYCLES FORCED BY pk ´ 2q-DEGREE
7
Proof of Corollary 8. For (a ), the second part of Lemma 7 for L and L1 implies that they
`|Bε |´ℓ˘
pk´ℓ´1q
both extend to an edge with at least p1 ´ 2δq|Aε | k´ℓ´1
sets in Ap1q
. Only few of
ε Bε
pk´ℓ´1q
those intersect U and by an averaging argument we obtain two sets C, C 1 P Ap1q
ε Bε
such that |C XC 1 | “ ℓ and LYC as well as L1 YC 1 are edges in H, which yields the required
ℓ-path. In view of (6), (b ) is a trivial consequence of the second part of Lemma 7.
Proof of Lemma 7. Let m ď k ´ 2 and let M P Bεpmq be an m-set. We will make use of
the following sum over all pk ´ 2q-sets D Ă Bε that contain M.
ÿ
ÿ ´
dpD, Aεp1q Bεp1q q ` dpD, pAε Y Vε qp2q q
dpDq “
M ĂDĂBε
|D|“k´2
M ĂDĂBε
|D|“k´2
` dpD, Bεp2q q ` dpD, Vεp1q Bεp1q q
Note that we can relate the sums
in question as follows.
ř
dpD, Aεp1q Bεp1q q and
pk´m´1q
dpM, Ap1q
q“
ε Bε
ř
(7)
¯
dpD, Bεp2q q in (7) to the terms
ÿ
1
dpD, Aεp1q Bεp1q q,
k ´ m ´ 1 M ĂDĂBε
|D|“k´2
dpM, Bεpk´mq q
1
“ `k´m˘
2
ÿ
M ĂDĂBε
|D|“k´2
(8)
dpD, Bεp2q q.
We will bound some of the terms on the right-hand side of (7). It directly follows from (5)
`
˘
that dpD, pAε Y Vε qp2q q ď a`3ϑb
; moreover, dpD, Vεp1q Bεp1q q ď 2ϑbbε . Using the minimum
2
pk ´ 2q-degree condition (3) we obtain
˙
˙ ˆˆ ˙
ˆ
ÿ
a
bε ´ m
` apb ´ k ` 2q .
dpDq ą
2
k´m´2
M ĂDĂBε
|D|“k´2
Combining these estimates with (7) and (8) yields
k´m
dpM, Aεp1q Bεpk´m´1q q `
dpM, Bεpk´mq q
2
˙
˙
ˆ
˙ ˆˆ ˙
ˆ
a ` 3ϑb
a
bε ´ m
1
´ 2ϑbbε
` apb ´ k ` 2q ´
ě
2
2
k´m´1 k´m´2
˙
ˆ
bε ´ m
.
ě p1 ´ δq aε
k´m´1
For the second part of the lemma, note that the definition of ε-typicality and ε ! δ imply
` bε ´ℓ ˘
that k´ℓ
dpL, Bεpk´ℓq q is smaller than δaε k´ℓ´1
for any ε-typical ℓ-set L, which concludes
2
the proof.
8
J. DE O. BASTOS, G. O. MOTA, M. SCHACHT, J. SCHNITZER, AND F. SCHULENBURG
For Lemma 5, we want to construct an ℓ-path Q, such that Vε Ă V pQq and the remaining
sets Aε r Q and Bε r Q have the right relative proportion of vertices, i.e., their sizes are in
a ratio of one to p2k ´ 2ℓ ´ 1q. If |AX Bε | ą 0, then B Ă Bε (see (4)) and so Q should cover
Vε and contain the right number of vertices from Bε . For this, we have to find suitable
edges inside Bε , which the following lemma ensures.
Lemma 9. Suppose that q “ |A X Bε | ą 0. Then there exist 2q ` 2 disjoint paths of size
three, each of which contains exactly one vertex from Aε and has two ε-typical sets as its
ends.
Proof. We say that an pℓ ´ 1q-set M Ă Bε is good if it is a subset of at least p1 ´
?
ε1 qbε
ε-typical sets, otherwise we say that the set is bad. We will first show that there are 2q ` 2
edges in Bε , each containing one ε-typical and one good pℓ ´ 1q-set. Then we will connect
pairs of these edges to ℓ-paths of size three.
Suppose that q “ |A X Bε | ą 0. So B Ă Bε by (4) and consequently |Bε | “ |B| ` q and
?
q ď ϑ|B|. It is not hard to see from (6) that at most a ε1 fraction of the pℓ ´ 1q-sets
in Bεpl´1q are bad. Hence, at least
˙
˙
ˆ
ˆ
ˆ
˙
˙ˆ
k´2 ? 1
k´2 1
b
ε ´
1´
ε
ℓ´1
ℓ
k´2
pk ´ 2q-sets in Bε contain no ε-atypical or bad subset. Let B Ă Bεpkq be the set of edges
inside Bε that contain such a pk ´ 2q-set. For all M P Bεpk´2q , by the minimum degree
`˘
condition, we have dpM, Bεp2q q ě qpb ´ k ` 2q ` q2 and, with the above, we have
˙
˙ˆ
˙
˙
ˆ
ˆ
ˆ
qpb ´ k ` 2q
b
k´2 ? 1
k´2 1
`k ˘
ε
ε ´
|B| ě 1 ´
k´2
ℓ´1
ℓ
2
˙
˙ˆ
˙
˙
ˆ
ˆ
ˆ
˙
ˆ
?
2q
b
k´2
k´2 1
q
b
1
ε ´
“ 1´
.
ě
ε
k´1 k
ℓ´1
ℓ
k k´1
` bε ˘
On the other hand, for any v P Bε we have dpv, Bεpk´1q q ă 2ε k´1
which implies that any
` bε ˘
edge in B intersects at most 2kε k´1 other edges in B. So, in view of ε ! k1 we may pick
a set B1 of 2q ` 2 disjoint edges in B.
We will connect each of the edges in B1 to an ε-typical set. Assume we have picked the
first i´1 desired ℓ-paths, say P1 , . . . , Pi´1 , and denote by U the set of vertices contained in
one of the paths or one of the edges in B1 . For the rest of this proof, when we pick vertices
and edges, they shall always be disjoint from U and everything chosen before. Let e be
an edge in B1 we have not considered yet and pick an arbitrary ε-typical set L1 Ă Bε r U.
We will first handle the cases that 2ℓ ` 1 ă k or that ℓ “ 1, k “ 3. In the first case,
a pk ´ 2q-set that contains no ε-atypical set already contains two disjoint ε-typical sets.
In the second case, an ℓ-set tvu is ε-typical for any vertex v in Bε by the definition of
LOOSE HAMILTONIAN CYCLES FORCED BY pk ´ 2q-DEGREE
9
ε-typicality. Hence in both cases e contains two disjoint ε-typical sets, say L0 and L1 . We
can use Corollary 8 (a ), as |U| ď 6kq, to connect L1 to L1 and obtain an ℓ-path Pi of size
three that contains one vertex in Aε and has ε-typical ends L0 and L1 .
So now assume that 2ℓ ` 1 “ k and k ą 3, in particular k ´ 2 “ 2ℓ ´ 1 and we may
split the pk ´ 2q-set considered in the definition of B into an ε-typical ℓ-set L and a good
pℓ ´ 1q-set G. Moreover, let w P e r pL Y Gq be one of the remaining two vertices and
set N “ G Y w.
δ
pℓq
First assume that dpN, Ap1q
ε Bε q ě 3 aε
`bε ˘
`˘
pℓq
. As ϑ ! δ, at most 3δ aε bℓ sets in Ap1q
ε Bε
ℓ
pℓq
1
intersect U. So it follows from Lemma 7 that there exist Ap1q
ε Bε -sets C, C in Bε such
that N Y C and L1 Y C 1 are edges, |C X C 1 | “ ℓ and C X C 1 X Aε ‰ ∅.
`bε ˘
δ
pℓq
Now assume that dpN, Ap1q
ε Bε q ă 3 aε ℓ . As the good set G forms an ε-typical set
with most vertices in Bε , there exists v P Bε r U such that
˙
ˆ
bε
p1q pk´ℓ´2q
dpN Y tvu, Aε Bε
q ă δaε
ℓ´1
and G Y tvu is an ε-typical set. Lemma 7 implies that
˙
ˆ ˙
ˆ
bε
bε
2
pℓq
.
ěδ
dpN Y tvu, Bε q ě p1 ´ 2δqaε
ℓ
ℓ
ℓ´1
So there exists an ε-typical ℓ-set L˚ Ă pBε r Uq such that N Y L˚ Y tvu is an edge in H.
Use Lemma 8 (a ) to connect L˚ to L1 and obtain an ℓ-path Pi of size three that contains
one vertex in Aε and has ε-typical ends G Y tvu and L1 .
If the hypergraph we consider is very close to the extremal example then Lemma 9 does
not apply and we will need the following lemma.
Lemma 10. Suppose that B “ Bε . If n is an odd multiple of k ´ ℓ then there exists a
single edge on Bε containing two ε-typical ℓ-sets. If n is an even multiple of k ´ ℓ then
there either exist two disjoint edges on Bε each containing two ε-typical ℓ-sets or an ℓ-path
of size two with ε-typical ends.
Proof. For the proof of this lemma all vertices and edges we consider will always be completely contained in Bε . First assume that there exists an ε-atypical ℓ-set L. Recall that
` |B| ˘
this means that dpL, B pk´ℓq q ą ε k´ℓ
so in view of (6) and ε1 ! ε we can find two disjoint
pk ´ ℓq-sets extending it to an edge, each containing an ε-typical set, which would prove
the lemma.
So we may assume that all ℓ-sets in Bεpℓq are ε-typical. We infer from the minimum
degree condition that Bε contains a single edge, which proves the lemma in the case that
n is an odd multiple of k ´ ℓ and for the rest of the proof we assume that n is an even
multiple of k ´ ℓ.
10
J. DE O. BASTOS, G. O. MOTA, M. SCHACHT, J. SCHNITZER, AND F. SCHULENBURG
Assume for a moment that ℓ “ 1. Recall that in this case any pk ´ 2q-set in B in the
extremal hypegraph Xk,ℓ pnq is contained in one edge. Consequently, the minimum degree
condition implies that any pk ´ 2q-set in Bε extends to at least two edges on Bε . Fix some
edge e in Bε ; any other edge on Bε has to intersect e in at least two vertices or the lemma
would hold. Consider any pair of disjoint pk ´ 2q-sets K and M in Bε r e to see that of
the four edges they extend to, there is a pair which is either disjoint or intersect in one
vertex, proving the lemma for the case ℓ “ 1.
Now assume that ℓ ą 1. In this case the minimum degree condition implies that any
k ´ 2-set in Bε extends to at least one edge on Bε . Again, fix some edge e in Bε ; any other
edge on Bε has to intersect e in at least one vertex or the lemma would hold. Applying the
minimum degree condition to all pk ´2q-sets disjoint from e implies that one vertex v P e is
`|Bε |˘
contained in at least 2k12 k´2
edges. We now consider the pk ´ 1q-uniform link hypergraph
of v on Bε . Any two edges intersecting in ℓ ´ 1 vertices would finish the proof of the
lemma. However a result of Frankl and Füredi [3] on set systems with a single forbidden
intersection size implies that this pk ´ 1q-uniform hypergraph without an intersection of
` |Bε | ˘
size ℓ ´ 1 contains at most k´ℓ´1
edges, a contradiction.
The following lemma will allow us to handle the vertices in Vε .
Lemma 11. Let U Ă Bε with |U| ď 4kϑ. There exists a family P1 , . . . , Pvε of disjoint
ℓ-paths of size two, each of which is disjoint from U such that for all i P rvε s
|V pPi q X Vε | “ 1
and |V pPi q X Bε | “ 2k ´ ℓ ´ 1,
and both ends of Pi are ε-typical sets.
Proof. Let Vε “ tx1 , . . . , xvε u. We will iteratively pick the paths. Assume we have already
chosen ℓ-paths P1 , . . . , Pi´1 containing the vertices v1 , . . . , vi´1 and satisfying the lemma.
Let U 1 be the set of all vertices in U or in one of those ℓ-paths. From vi R Bε we get
dpvi , Bεpk´1q q
ˆ
|B|
ě dpvi , Bq ´ |B r Bε | ¨
k´2
˙
˙
ˆ
bε
ε
.
ě
2 k´1
` bε ˘
From (6) we get that at most k ℓ ε1 k´1
sets in Bεpk´1q contain at least one ε-atypical ℓ-set.
`
˘
bε
Also, less than 8ε k´1
sets in Bεpk´1q contain one of the vertices of U 1 . In total, at least
`
˘
ε bε
of the Bεpk´1q -sets form an edge with vi . So we may pick two edges e and f in
4 k´1
Vεp1q Bεpk´1q that contain the vertex vi and intersect in ℓ vertices. In particular, these edges
form an ℓ-path of size two as required by the lemma.
LOOSE HAMILTONIAN CYCLES FORCED BY pk ´ 2q-DEGREE
11
We can now proceed with the proof of Lemma 5. Recall that we want to prove the
existence of an ℓ-path Q in H with ends L0 and L1 and a partition
A˚ Ÿ B˚ “ pV r Qq Ÿ L0 Ÿ L1
satisfying properties (i )–(iv ) of Lemma 5. Set q “ |A X Bε |. We will split the construction
of the ℓ-path Q into two cases, depending on whether q “ 0 or not.
First, suppose that q ą 0. In the following, we denote by U the set of vertices of all
edges and ℓ-paths chosen so far. Note that we will always have |U| ď 20kϑn and hence we
will be in position to apply Corollary 8. We use Lemma 9 to obtain paths Q1 , . . . , Q2q`2
and then we apply Lemma 11 to obtain ℓ-paths P1 , . . . , Pvε . Every path Qi , for i P r2q `2s,
contains 3k ´ 2ℓ ´ 1 vertices from Bε and one from Aε , while every Pj , for j P rvε s, contains
2k ´ ℓ ´ 1 from Bε and one from Vε .
As the ends of all these paths are ε-typical, we apply Corollary 8 (a ) repeatedly to
connect them to one ℓ-path P. In each of the vε ` 2q ` 1 steps of connecting two ℓ-paths,
we used one vertex from Aε and 2k ´ 3ℓ ´ 1 vertices from Bε . Overall, we have that
|V pPq X Aε | “ vε ` 4q ` 3,
as well as
|V pPq X Bε | “ p4k ´ 4ℓ ´ 2qvε ` p5k ´ 5ℓ ´ 2qp2q ` 2q ´ p2k ´ 3ℓ ´ 1q.
Furthermore |V pPq| ď 10kϑb.
Using the identities aε `bε `vε “ n and aε `q`vε “ a, we will now establish property (i )
of Lemma 5. Set spPq “ p2k ´ 2ℓ ´ 1q|Aε r V pPq| ´ |Bε r V pPq| ´ 2ℓ, so
spPq “ p2k ´ 2ℓ ´ 1q|Aε r V pPq| ´ |Bε r V pPq| ´ 2ℓ
“ p2k ´ 2ℓ ´ 1qpaε ´ pvε ` 4q ` 3qq ´ bε
` p4k ´ 4ℓ ´ 2qvε ` p5k ´ 5ℓ ´ 2qp2q ` 2q ´ p2k ´ 3ℓ ´ 1q ´ 2ℓ
“ p2k ´ 2ℓ ´ 1qaε ´ bε ` p2k ´ 2ℓ ´ 1qvε ` 2pk ´ ℓqq ` 2k ´ 3ℓ
“ 2pk ´ ℓqpaε ` vε ` q ` 1q ´ n ´ ℓ
“ 2pk ´ ℓqpa ` 1q ´ n ´ ℓ.
If n{pk ´ ℓq is even, spPq “ ´ℓ (see (2)) and we set Q “ P. Otherwise spPq “ k ´ 2ℓ
and we use Corollary 8 (b ) to append one edge to P to obtain Q. It is easy to see that
one application of Corollary 8 (b ) decreases spPq by k ´ ℓ. Setting A˚ “ Aε r V pQq and
B˚ “ pBε r V pQqq Y L0 Y L1 we get from spQq “ ´ℓ that A˚ and B˚ satisfy (i ).
12
J. DE O. BASTOS, G. O. MOTA, M. SCHACHT, J. SCHNITZER, AND F. SCHULENBURG
Now, suppose that q “ 0. Apply Claim 11 to obtain ℓ-paths P1 , . . . , Pvε . If B “
Bε , apply Lemma 10 to obtain one or two more ℓ-paths contained in Bε . We apply
Corollary 8 (a ) repeatedly to connect them to one ℓ-path P.
Since q “ 0, we have that Bε Ă B and aε ` vε “ |V r Bε | “ a ` |B r Bε |. We
can assume without loss of generality that Vε ‰ ∅, otherwise just take Vε “ tvu for an
arbitrary v P V pHq. If B “ Bε let x be 2pk ´ ℓq or k ´ ℓ depending on whether n is an
odd or even multiple of k ´ ℓ; otherwise let x “ 0. With similar calculations as before and
the same definition of spPq we get that
spPq “ 2pk ´ ℓqa ` x ` 2pk ´ ℓq|B r Bε | ´ n ´ ℓ ” ´ℓ
Extend the ℓ-path P to an ℓ-path Q by adding
spPq`ℓ
k´l
mod pk ´ ℓq.
edges using Corollary 8 (b ). Thus
spQq “ ´ℓ, and we get (i ) as in the previous case.
In both cases, we will now use the properties of the constructed ℓ-path Q to show (ii )-
(iv ). We will use that vpQq ď 20kϑb, which follows from the construction. Since A˚ Ă Aε ,
for all v P A˚ we have dpv, B pk´1q q ě p1 ´ εqB pk´1q . Thus
˙
˙
ˆ
ˆ
|B˚ |
|B˚ | ´ 1
pk´1q
pk´1q
,
ě p1 ´ 2εq
dpv, B˚
q ě dpv, B
q ´ |B˚ r B|
k´1
k´2
which shows (ii ).
For (iii ), Lemma 7 yields for all vertices v P B˚ Ă Bε that
˙
ˆ
k´1
|Bε | ´ 1
pk´1q
`
.
dpv, Bε
q ě p1 ´ δq |Aε |
2
k´2
` bε ˘
The second term on the left can be bounded from above by 2kε k´1
. So, as δ, ε ! ̺ and
pk´2q
dpv, Ap1q
q
ε Bε
aε ´ |A˚ | ! ̺|A˚ | as well as bε ´ |B˚ | ! ̺|B˚ |, we can conclude (iii ).
By Lemma 7, we know that
pk´1q
dpL0 , Ap1q
q, dpL1 , Aεp1q Bεpk´1q q
ε Bε
ě p1 ´ δqaε
˙
bε ´ ℓ
.
k´ℓ´1
ˆ
As δ ! ̺ and aε ´ |A˚ | ! ̺|A˚ | as well as bε ´ |B˚ | ! ̺|B˚ |, we can conclude (iv ).
References
[1] J. de O. Bastos, G. O. Mota, M. Schacht, J. Schnitzer, and F. Schulenburg, Loose Hamiltonian cycles
forced by large (k-2)-degree – approximate version. Submitted. Ò2, 3
[2] E. Buß, H. Hàn, and M. Schacht, Minimum vertex degree conditions for loose Hamilton
cycles in 3-uniform hypergraphs, J. Combin. Theory Ser. B 103 (2013), no. 6, 658–678,
DOI 10.1016/j.jctb.2013.07.004. MR3127586 Ò2
[3] P. Frankl and Z. Füredi, Forbidding just one intersection, J. Combin. Theory Ser. A 39 (1985), no. 2,
160–176, DOI 10.1016/0097-3165(85)90035-4. MR793269 Ò10
LOOSE HAMILTONIAN CYCLES FORCED BY pk ´ 2q-DEGREE
13
[4] H. Hàn and M. Schacht, Dirac-type results for loose Hamilton cycles in uniform hypergraphs, J.
Combin. Theory Ser. B 100 (2010), no. 3, 332–346, DOI 10.1016/j.jctb.2009.10.002. MR2595675 Ò2
[5] J. Han and Y. Zhao, Minimum codegree threshold for Hamilton ℓ-cycles in k-uniform hypergraphs, J.
Combin. Theory Ser. A 132 (2015), 194–223, DOI 10.1016/j.jcta.2015.01.004. MR3311344 Ò4
[6]
, Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs, J.
Combin. Theory Ser. B 114 (2015), 70–96, DOI 10.1016/j.jctb.2015.03.007. MR3354291 Ò2
[7] G. Y. Katona and H. A. Kierstead, Hamiltonian chains in hypergraphs, J. Graph Theory
30 (1999), no. 3, 205–212, DOI 10.1002/(SICI)1097-0118(199903)30:3<205::AID-JGT5>3.3.CO;2-F.
MR1671170 Ò1
[8] P. Keevash, D. Kühn, R. Mycroft, and D. Osthus, Loose Hamilton cycles in hypergraphs, Discrete
Math. 311 (2011), no. 7, 544–559, DOI 10.1016/j.disc.2010.11.013. MR2765622 Ò2
[9] D. Kühn, R. Mycroft, and D. Osthus, Hamilton ℓ-cycles in uniform hypergraphs, J. Combin. Theory
Ser. A 117 (2010), no. 7, 910–927, DOI 10.1016/j.jcta.2010.02.010. MR2652102 Ò2
[10] D. Kühn and D. Osthus, Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree, J.
Combin. Theory Ser. B 96 (2006), no. 6, 767–821, DOI 10.1016/j.jctb.2006.02.004. MR2274077 Ò2
[11] V. Rödl and A. Ruciński, Dirac-type questions for hypergraphs—a survey (or more problems for Endre
to solve), An irregular mind, Bolyai Soc. Math. Stud., vol. 21, János Bolyai Math. Soc., Budapest,
2010, pp. 561–590, DOI 10.1007/978-3-642-14444-8_16. MR2815614 Ò1
[12] V. Rödl, A. Ruciński, and E. Szemerédi, A Dirac-type theorem for 3-uniform hypergraphs, Combin.
Probab. Comput. 15 (2006), no. 1-2, 229–251, DOI 10.1017/S0963548305007042. MR2195584 Ò1
[13]
, An approximate Dirac-type theorem for k-uniform hypergraphs, Combinatorica 28 (2008),
no. 2, 229–260, DOI 10.1007/s00493-008-2295-z. MR2399020 Ò1
[14] Y. Zhao, Recent advances on Dirac-type problems for hypergraphs, Recent trends in combinatorics,
IMA Vol. Math. Appl., vol. 159, Springer, 2016, pp. 145–165, DOI 10.1007/978-3-319-24298-9_6.
MR3526407 Ò1
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil
E-mail address: {josefran|mota}@ime.usp.br
Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
E-mail address: [email protected]
E-mail address: {jakob.schnitzer|fabian.schulenburg}@uni-hamburg.de